# Properties

 Label 2592.2.r.p Level 2592 Weight 2 Character orbit 2592.r Analytic conductor 20.697 Analytic rank 0 Dimension 8 CM no Inner twists 8

# Related objects

## Newspace parameters

 Level: $$N$$ = $$2592 = 2^{5} \cdot 3^{4}$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 2592.r (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$20.6972242039$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: 8.0.49787136.1 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{8}$$ Twist minimal: no (minimal twist has level 216) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{4} q^{5} + ( 1 - \beta_{3} ) q^{7} +O(q^{10})$$ $$q + \beta_{4} q^{5} + ( 1 - \beta_{3} ) q^{7} -3 \beta_{1} q^{11} -\beta_{2} q^{13} + ( -\beta_{5} + \beta_{7} ) q^{17} + \beta_{6} q^{19} -\beta_{5} q^{23} + ( -4 + 4 \beta_{3} ) q^{25} + 6 \beta_{1} q^{29} -7 \beta_{3} q^{31} + ( \beta_{1} + \beta_{4} ) q^{35} + \beta_{6} q^{37} -\beta_{5} q^{41} + ( 2 \beta_{2} - 2 \beta_{6} ) q^{43} + 6 \beta_{3} q^{49} + ( 9 \beta_{1} + 9 \beta_{4} ) q^{53} + 3 q^{55} + 4 \beta_{4} q^{59} + \beta_{7} q^{65} + ( -3 \beta_{5} + 3 \beta_{7} ) q^{71} + 3 q^{73} + 3 \beta_{4} q^{77} + ( 4 - 4 \beta_{3} ) q^{79} + 7 \beta_{1} q^{83} -\beta_{2} q^{85} + ( -2 \beta_{5} + 2 \beta_{7} ) q^{89} -\beta_{6} q^{91} -\beta_{5} q^{95} + ( -7 + 7 \beta_{3} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 4q^{7} + O(q^{10})$$ $$8q + 4q^{7} - 16q^{25} - 28q^{31} + 24q^{49} + 24q^{55} + 24q^{73} + 16q^{79} - 28q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 3 x^{6} + 5 x^{4} + 12 x^{2} + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{7} - 5 \nu^{5} + 5 \nu^{3} + 12 \nu$$$$)/40$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{6} + 15 \nu^{4} + 5 \nu^{2} + 12$$$$)/10$$ $$\beta_{3}$$ $$=$$ $$($$$$3 \nu^{6} + 5 \nu^{4} + 15 \nu^{2} + 36$$$$)/20$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{7} - 7 \nu$$$$)/10$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{7} + 13 \nu$$$$)/5$$ $$\beta_{6}$$ $$=$$ $$($$$$-4 \nu^{6} - 18$$$$)/5$$ $$\beta_{7}$$ $$=$$ $$($$$$11 \nu^{7} + 25 \nu^{5} + 55 \nu^{3} + 132 \nu$$$$)/20$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{5} - 2 \beta_{4}$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{6} + 6 \beta_{3} - \beta_{2} - 6$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{7} - \beta_{5} + 10 \beta_{4} + 10 \beta_{1}$$$$)/4$$ $$\nu^{4}$$ $$=$$ $$($$$$-2 \beta_{3} + 3 \beta_{2}$$$$)/4$$ $$\nu^{5}$$ $$=$$ $$($$$$\beta_{7} - 22 \beta_{1}$$$$)/4$$ $$\nu^{6}$$ $$=$$ $$($$$$-5 \beta_{6} - 18$$$$)/4$$ $$\nu^{7}$$ $$=$$ $$($$$$-7 \beta_{5} - 26 \beta_{4}$$$$)/4$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2592\mathbb{Z}\right)^\times$$.

 $$n$$ $$325$$ $$1217$$ $$2431$$ $$\chi(n)$$ $$-1$$ $$-\beta_{3}$$ $$1$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
433.1
 −0.228425 + 1.39564i 1.09445 − 0.895644i 0.228425 − 1.39564i −1.09445 + 0.895644i −0.228425 − 1.39564i 1.09445 + 0.895644i 0.228425 + 1.39564i −1.09445 − 0.895644i
0 0 0 −0.866025 0.500000i 0 0.500000 + 0.866025i 0 0 0
433.2 0 0 0 −0.866025 0.500000i 0 0.500000 + 0.866025i 0 0 0
433.3 0 0 0 0.866025 + 0.500000i 0 0.500000 + 0.866025i 0 0 0
433.4 0 0 0 0.866025 + 0.500000i 0 0.500000 + 0.866025i 0 0 0
2161.1 0 0 0 −0.866025 + 0.500000i 0 0.500000 0.866025i 0 0 0
2161.2 0 0 0 −0.866025 + 0.500000i 0 0.500000 0.866025i 0 0 0
2161.3 0 0 0 0.866025 0.500000i 0 0.500000 0.866025i 0 0 0
2161.4 0 0 0 0.866025 0.500000i 0 0.500000 0.866025i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 2161.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
8.b even 2 1 inner
9.c even 3 1 inner
9.d odd 6 1 inner
24.h odd 2 1 inner
72.j odd 6 1 inner
72.n even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2592.2.r.p 8
3.b odd 2 1 inner 2592.2.r.p 8
4.b odd 2 1 648.2.n.n 8
8.b even 2 1 inner 2592.2.r.p 8
8.d odd 2 1 648.2.n.n 8
9.c even 3 1 864.2.d.a 4
9.c even 3 1 inner 2592.2.r.p 8
9.d odd 6 1 864.2.d.a 4
9.d odd 6 1 inner 2592.2.r.p 8
12.b even 2 1 648.2.n.n 8
24.f even 2 1 648.2.n.n 8
24.h odd 2 1 inner 2592.2.r.p 8
36.f odd 6 1 216.2.d.b 4
36.f odd 6 1 648.2.n.n 8
36.h even 6 1 216.2.d.b 4
36.h even 6 1 648.2.n.n 8
72.j odd 6 1 864.2.d.a 4
72.j odd 6 1 inner 2592.2.r.p 8
72.l even 6 1 216.2.d.b 4
72.l even 6 1 648.2.n.n 8
72.n even 6 1 864.2.d.a 4
72.n even 6 1 inner 2592.2.r.p 8
72.p odd 6 1 216.2.d.b 4
72.p odd 6 1 648.2.n.n 8
144.u even 12 1 6912.2.a.bc 2
144.u even 12 1 6912.2.a.bu 2
144.v odd 12 1 6912.2.a.bc 2
144.v odd 12 1 6912.2.a.bu 2
144.w odd 12 1 6912.2.a.bd 2
144.w odd 12 1 6912.2.a.bv 2
144.x even 12 1 6912.2.a.bd 2
144.x even 12 1 6912.2.a.bv 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
216.2.d.b 4 36.f odd 6 1
216.2.d.b 4 36.h even 6 1
216.2.d.b 4 72.l even 6 1
216.2.d.b 4 72.p odd 6 1
648.2.n.n 8 4.b odd 2 1
648.2.n.n 8 8.d odd 2 1
648.2.n.n 8 12.b even 2 1
648.2.n.n 8 24.f even 2 1
648.2.n.n 8 36.f odd 6 1
648.2.n.n 8 36.h even 6 1
648.2.n.n 8 72.l even 6 1
648.2.n.n 8 72.p odd 6 1
864.2.d.a 4 9.c even 3 1
864.2.d.a 4 9.d odd 6 1
864.2.d.a 4 72.j odd 6 1
864.2.d.a 4 72.n even 6 1
2592.2.r.p 8 1.a even 1 1 trivial
2592.2.r.p 8 3.b odd 2 1 inner
2592.2.r.p 8 8.b even 2 1 inner
2592.2.r.p 8 9.c even 3 1 inner
2592.2.r.p 8 9.d odd 6 1 inner
2592.2.r.p 8 24.h odd 2 1 inner
2592.2.r.p 8 72.j odd 6 1 inner
2592.2.r.p 8 72.n even 6 1 inner
6912.2.a.bc 2 144.u even 12 1
6912.2.a.bc 2 144.v odd 12 1
6912.2.a.bd 2 144.w odd 12 1
6912.2.a.bd 2 144.x even 12 1
6912.2.a.bu 2 144.u even 12 1
6912.2.a.bu 2 144.v odd 12 1
6912.2.a.bv 2 144.w odd 12 1
6912.2.a.bv 2 144.x even 12 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(2592, [\chi])$$:

 $$T_{5}^{4} - T_{5}^{2} + 1$$ $$T_{13}^{4} - 28 T_{13}^{2} + 784$$ $$T_{17}^{2} - 28$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$( 1 + 9 T^{2} + 56 T^{4} + 225 T^{6} + 625 T^{8} )^{2}$$
$7$ $$( 1 - 5 T + 7 T^{2} )^{4}( 1 + 4 T + 7 T^{2} )^{4}$$
$11$ $$( 1 + 13 T^{2} + 48 T^{4} + 1573 T^{6} + 14641 T^{8} )^{2}$$
$13$ $$( 1 - 2 T^{2} - 165 T^{4} - 338 T^{6} + 28561 T^{8} )^{2}$$
$17$ $$( 1 + 6 T^{2} + 289 T^{4} )^{4}$$
$19$ $$( 1 - 10 T^{2} + 361 T^{4} )^{4}$$
$23$ $$( 1 - 18 T^{2} - 205 T^{4} - 9522 T^{6} + 279841 T^{8} )^{2}$$
$29$ $$( 1 + 22 T^{2} - 357 T^{4} + 18502 T^{6} + 707281 T^{8} )^{2}$$
$31$ $$( 1 - 4 T + 31 T^{2} )^{4}( 1 + 11 T + 31 T^{2} )^{4}$$
$37$ $$( 1 - 46 T^{2} + 1369 T^{4} )^{4}$$
$41$ $$( 1 - 54 T^{2} + 1235 T^{4} - 90774 T^{6} + 2825761 T^{8} )^{2}$$
$43$ $$( 1 - 26 T^{2} - 1173 T^{4} - 48074 T^{6} + 3418801 T^{8} )^{2}$$
$47$ $$( 1 - 47 T^{2} + 2209 T^{4} )^{4}$$
$53$ $$( 1 - 25 T^{2} + 2809 T^{4} )^{4}$$
$59$ $$( 1 + 102 T^{2} + 6923 T^{4} + 355062 T^{6} + 12117361 T^{8} )^{2}$$
$61$ $$( 1 + 61 T^{2} + 3721 T^{4} )^{4}$$
$67$ $$( 1 + 67 T^{2} + 4489 T^{4} )^{4}$$
$71$ $$( 1 - 110 T^{2} + 5041 T^{4} )^{4}$$
$73$ $$( 1 - 3 T + 73 T^{2} )^{8}$$
$79$ $$( 1 - 17 T + 79 T^{2} )^{4}( 1 + 13 T + 79 T^{2} )^{4}$$
$83$ $$( 1 + 117 T^{2} + 6800 T^{4} + 806013 T^{6} + 47458321 T^{8} )^{2}$$
$89$ $$( 1 + 66 T^{2} + 7921 T^{4} )^{4}$$
$97$ $$( 1 + 7 T - 48 T^{2} + 679 T^{3} + 9409 T^{4} )^{4}$$